1) Depends on time of day and the day. Sometimes, I'm waiting really long and sometimes the invite is nearly instant. But on average you will wait a long time.
2) Spec is all that matters for the loot (that must be since druids and shamans have two dps specs that need totally different gear). So queueing as dps and staying in tank spec will only flatten your damage and so your contribution to the raid
3) see 2)

If I understand some of you correctly, a Protection Warrior can queue up as DPS in *Protection* spec and get tank gear?

Also are you allowed to switch spec once you are in the raid? If yes, how will that affect loot please?

You can sign up as a DPS in the Raid Finder UI, but stay in Tank spec. This will mean you get Tank loot, not DPS loot. What you sign up as on the UI has absolutely no relevance to what loot you get: it is only the spec you are that counts.

Yes, you can change spec when in the raid and change per boss. This will then change what loot you get: see above paragraph. You do not get 'locked' to a certain group of loot for an entire raid. You can be DPS for the first boss, tank for the second, then heals for the third. It's on an individual, per boss basis.

Your current talent spec is what determines your loot, not what you queue as. So you can queue as DPS and then switch to your tank spec for a fight to get tank gear, or vice-versa for 1-tank fights like Spirit Kings.

This is frowned upon though. As a prot-warrior will do "much" less damage than a dps-warrior. Then again, it's unlikely anyone will find out.

In Heroic dungeons tanks usually have instaqueue. LFR shouldn't be much worse.

Quick maths lesson.

In dungeons tanks make up 20% of a group. In lfr they are 8% of a group. Healers make up 24% and dps the rest. Queue times are therefore quickest for healers in lfr, then dps and tanks have by far the longest.